Faulty modem? You have been pretty thorough.
A cheap-but-excellent DLink DSL-320B-Z1 modem - just works. (ADSLx only - Not for VDSL2.)
It is essential for security and sanity to configure it as a straight modem-only device, not as a modem-router. This is because it's full of nasty bugs and security holes if allowed to be a router.
Configuring it properly is very easy, we can help. If you buy a cheap one from the Andrews & Arnold shop they come configured safely for you, so that saves a small job.
http://aa.net.uk/broadband-accessories.html (AA Shop)
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Important: however, iirc, as AA ship them (possibly by default) the bitswap enable tick box is unticked.
It's essential that you tick this for decent performance and reliability, unless this is just a bug in the UI, but I notice that user G.DMT tested it. I complained to AA about this iirc but they didn't take any notice of me.
The Draytek 130 is an excellent modem if you have a short line. Not for long lines. Just plug in, zero config for me. I forget which ISP you're using ?